A C⁰-nonconforming quadrilateral finite element for the fourth-order elliptic singular perturbation problem

Bao, Yuan; Meng, Zhaoliang; Luo, Zhongxuan

ESAIM: Mathematical Modelling and Numerical Analysis , Tome 52 (2018) no. 5, pp. 1981-2001.

In this paper, a $C^0$ nonconforming quadrilateral element is proposed to solve the fourth-order elliptic singular perturbation problem. For each convex quadrilateral $Q$, the shape function space is the union of $S_2^1\left(Q^{*}\right)$ and a bubble space. The degrees of freedom are defined by the values at vertices and midpoints on the edges, and the mean values of integrals of normal derivatives over edges. The local basis functions of our element can be expressed explicitly by a new reference quadrilateral rather than by solving a linear system. It is shown that the method converges uniformly in the perturbation parameter. Lastly, numerical tests verify the convergence analysis.

Classification : 65N30
Mots clés : Singular perturbation problem, quadrilateral element, uniformly convergent

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